### Stocks are not Well Behaved

June 17, 2012
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So I stole this idea from Benoit Mandelbrot from his book *The Misbehavior of Markets* (which is an interesting book btw). I looked at daily % change in the price of the S&P 500 going back to 1950 (which is where yahoo finance's data begins). Here's what it looks like:

So the question to ask this: what would it look like if the volatility of day-to-day % change price movements look like if the returns were normally distributed. To do this, I used Random.org's Gaussian Random Number Generator to generate a list of random numbers comparable to the distribution for S&P 500 % price change. What I get looks like this:

Some general statistics:

Both had 15715 observations.

The Mean was 0.03% for the S&P 500 and 0.04% for the Normal Distribution Data.

Both had Standard Deviation of 0.98%

The Max/Min for the Normal Distribution Data were 3.00%/-2.90%.

The Max/Min for the S&P 500 Data were 11.58%/ -20.47%

As expected, the normal distribution stayed within about 3 standard deviations of the mean which should account for about 99.7% of the data. Contrast that with the S&P 500 daily changes and we find many points going outside of 3 standard deviations.

The fact that the volatility for stocks is not normally distributed has important implications. As an example, the Black-Scholes model for option pricing assumes that volatility follows a normal distribution. That's why option implied volatility shows a volatility smile, accounting for the fatter tails.